Course grades. Computing Your Course Grade and GPA, or, Weighted Averages

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1 Computing Your Course Grade and GPA, or, Weighted Averages The weighted average is one of the most common and useful formulas completely passed over introductory statistics classes. We are all familiar with the sample mean, x, where we add all the observations together then divide by the sample size: x 1 n Sometimes, though, we want to give certain values more inuence in determining the average. i1 Course grades A perfect example of this is when professors calculate your grade. Most courses have multiple assignments, and usually the assignments are worth dierent amounts. For example, a single homework assignment is not worth as much as the nal exam. One way to grade a course in this way is to make assignments worth dierent amount of points based on how important they are in the course grade calculation. If you want homework assignments to be worth 10% of what the nal is worth, you make the homework assignments 10 points a piece and the nal exam 100. Then you simply add all the points received during the quarter and divide by the total number of possible points. This starts to get complicated, though, if you have many assignments with dierent inuences. A second way to calculate grades is to allow every assignment to be graded out of any number of points, then weight the percentage received for each assignment in the correct manner. Suppose you are taking a class and the syllabus has this grade breakdown: Assignment Weight Homework 10% Midterm 20% Final Project 40% Final Exam 30% It's clear here that the nal project is a much more inuential part of your grade than the homework. Now suppose you received the following grades for each category: Assignment Weight Percentage Homework 10% 80% Midterm 20% 75% Final Project 40% 90% Final Exam 30% 85% To compute your course grade, you need to factor in the inuences of each assignment. This is done by converting the weights from percentages to proportions (10% becomes 0.10, 20% becomes 0.20, etc.), then multiplying your percentage for each category by the category weight and summing: course grade 0.10(80) (75) (90) (85)

2 So your overall grade is 84.5%. Notice that had you taken the simple sample mean of your category grades, you would have concluded your grade was 82.5: x 1 n i Your weighted grade is higher than the simple average because the grades on the nal assignments, which had more inuence, were better than the grades on the less inuential assignments. In this example, the calculations were simplied a bit because the weights summed to 1: What happens when the weights don't sum to 1? Let's look at your course grade going into the nal assignments, so you have only completed the homework and midterm. Computing your grade as before you would get 0.10(80) (75) or 23%, but this is clearly wrong - you received passing grades on both assignments! The 23% you see is actually 23% of the entire course grade, but you have only completed 30% of the work. In order to determine your actual grade, you need to divide by 0.30 (30%): or about 77%. This makes a lot more sense General formula for weighted averages In the course grade example above, the percentages received in each category are considered the data, or the x values, and the inuences (0.10, 0.20, 0.40, and 0.30) are called the weights. To nd the weighted average of a set of observations, multiply each observation by its weight, add the products together, then divide by the sum of the weights: x W i1 w i i1 w i In the rst example, the weights added to 1, or i1 w i 1, so the formula became simplied: x W w i i1 In determining which values are the observations ( ) and which are the weights (w i ), ask yourself, What do I want the average of? The answer to this question is your set of observations. GPA GPAs, or grade point averages, are also weighted averages. In calculating a GPA, the course grade (usually on a scale) is the observation, and the weight is the number of units or credits associated with the 2

3 course. Take the following grades from a student's quarter: Course Units Grade Intro Statistics Psychology Biology I To nd the student's GPA for the quarter, we need to use the units as the weights (w i ) and the grade as the observation ( ): GP A x W i1 w i i1 w i We could also expand the table to aid in our calculations: 4.0(3.7) + 3.0(4.0) + 5.0(3.0) Course Units Grade U nits Grade Intro Statistics Psychology Biology I Total: GPA: 41.8 / Again, this is a dierent result than we would have obtained by taking the sample average: x The actual GPA is lower because the student received lower grades in the courses worth more units. We can also use weighed averages to compute cumulative GPAs without going down to the level of individual courses. Suppose that before starting the quarter above, the student had a cumulative GPA of 3.1 after taking a total of 48 units. We create the following table: Time Units GPA Cumulative Current quarter Again, we use the units as weights. Now, though, the GPAs are the observation: 48(3.1) + 12(3.48) cumulative GP A The student's cumulative GPA has gone up, but it is not right in the middle of the previous GPA and this quarter's GPA because this quarter's GPA is based o of one fourth the units of the previous cumulative calculation. 3

4 Investment portfolios Weighted averages aren't jused used in computing grades. Another use of weighted averages are in investment portfolios. When people or corporations invest in the stock or bond market, they invest dierent amounts of money in various securities. Many websites will show investments as a pie chart. Suppose you invested in ve companies, with amounts given below: Company Amount Invested AT&T $5,000 Apple $4,000 Google $3,500 Starbucks $3,500 Ford $1,000 A pie chart of your portfolio would look like this: Apple AT&T Ford Google Starbucks Over the course of the year, you nd that AT&T went up 2%, Apple went up 10%, Google went down 4%, Starbucks went up 15%, and Ford went down 8%. How did your portfolio do overall? We want to nd the average change in our portfolio, so the individual stock changes are our observations, and the amount invested are the weights: Company Amount Invested (w i ) Change ( ) AT&T $5,000 2.% Apple $4, % Google $3,500-4.% Starbucks $3, % Ford $1,000-8.% Now the total change for the portfolio is x W i1 w i i1 w i 5000(2) (10) ( 4) (15) ( 8) % 4

5 So you have gained 4.7% on your total investments. For comparison, taking the sample average of the changes gives 3.0. The weighted average is higher becuase the biggest loss came from the company in which you had the smallest investment, so it had a relatively small eect on your portfolio. Weighted average and sample mean The common sample mean x 1 n xi is actually a special case of a weighted average where all observations are given the same weight. Let's say we have n observations and we give each the weight w i 1. Then the weighted average is x W i1 w i i1 w i i1 1 i1 1 1 n i i1 n x i1 You can check for yourself that there is nothing special about setting w i 1. Any weight will work as long as its the same for all observations since the weights in the numerator will cancel with the weights in the denominator. Weighted averages can also be used in place of sample means to make problems quicker when the observations only have a few unique values. Suppose we take a sample of 10 adults and ask how many pets they have at home. The responses are: To compute the sample mean, we take x 1 n i1 Alternatively, we can create a frequency table: 0, 0, 1, 1, 1 1, 1, 2 2, 2 1 ( ) Number of pets Frequency Then compute the weighted average using the frequencies as weights: x W 2(0) + 5(1) + 3(2)

6 We get the same answer. This occurs because in the weighted average, we are combining all the observations with the same number together. Lets compute x dierently: x 1 n i1 1 ( ) 10 1 [2(0) + 5(1) + 3(2)] 10 We see this is exactly the same formula as the weighted average. Generally speaking, when observations can be collected into sets with the same value, we can use the frequency of each set as weights to compute a weighted average and avoid long summations. 6

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